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H: Invertibility of row and column operations
I have a problem and a proposed plan for a solution. Please tell me if I'm on the right track.
Problem: What happens if instead of $1$ row operation and then $1$ column operation, the reverse order is performed on a matrix?
I'm thinking: There are $3$ types of row operatio... |
H: Integration By Parts with a definite integral
I've got the following:
\begin{align}
\int_{0}^{1}\int_{0}^{y^{2}}\frac{y}{x^{2}+y^{2}}\ dx\ dy&=\int_{0}^{1}\left.\arctan{\left(\frac{x}{y}\right)}\right|_{x=0}^{x=y^{2}}\ dy\\
&=\int_{0}^{1}\arctan{(y)}\ dy\\
&=y\arctan{(y)}-\int_{0}^{1}\frac{y}{1+y^{2}}\ dy
\end{alig... |
H: Integrating trig functions with $R(\frac {z+1/z} {2}, \frac {z - 1/z} {2i} )$
Someone told me that there is a method for integrating rational functions $R(\cos{\theta}, \sin { \theta})$ by doing contour integration of the complex function $$\frac {R \left( \frac {z + \frac1z} {2}, \frac {z - \frac1z} {2i} \right)} ... |
H: Proof of theorem about continuity
$\textbf{4.2}\,\,$ Theorem $\,\,$ Let $X,Y,E,f$, and $p$ be as in Definition $4.1$. Then
$$\lim_{x\to p}f(x)=q\tag{4}$$
if and only if
$$\lim_{n\to\infty}f(p_n)=q\tag{5}$$
for every sequence $\{p_n\}$ in $E$ such that
$$p_n\ne p,\quad\lim_{n\to\infty}p_n=p.\tag{6}$$
*Proof*$\quad$... |
H: what is difference between 0 and infinity norm?
Suppose $f$ is a real function on $\Omega$, both $\|f\|_\infty$ and $\|f\|_0$ are defined as
$\sup_{x\in \Omega} f(x)$ in many books. Then, am I missing some from their definitions?
AI: In the context that these are functions on a compact topological space $X$, the $... |
H: Proof for planar embeddings
Prove that any planar embedding of a simple connected planar graph contains a vertex of degree at most $3$ or a face of degree at most $3$.
Can someone help me with this please? Thank you!
AI: Hint: If every face has degree $4$ or more, then we can use Euler's polyhedron formula to show... |
H: How to show this line is tangent to $f$ at point $a$?
Let $f:I\to\mathbb{R}^n$ be a differentiable function, with $f'(a)\neq 0$ for some $a$ in the interval $I\subset\mathbb{R}$. If there exists a line $L\subset\mathbb{R}^n$ and a sequence $(x_k)$ in $I$such that $x_i\neq x_j$ when $i\neq j$, $\lim x_k=a$ and $f(x_... |
H: Why is it that $\{\vec{x}\}$ is always an orthogonal set?
Why is it that $\{\vec{x}\}$ is always an orthogonal set, assuming $\vec{x}\in\mathbb{R}^n$ and $\vec{x}\neq 0$?
AI: The set $\{\vec{x}\}$ is orthogonal because for any two distinct elements $\vec{v},\vec{w}\in\{\vec{x}\}$, we have
$\langle \vec{v},\vec{w}\r... |
H: On a bijection between symmetric subsets of a group
Given a group $G$, we can consider the subset $H$ of $G$ defined by:
$$ H = \{ xyz : x, y, z\in G \textrm{ and } x, y, z \textrm{ are pairwise distinct}\}$$
Let $a\in G$ be arbitrary element. I am interested in understanding the map
$f_a: H\to H$ defined by
$$
f_... |
H: Estimate a upper bound of an infinite series.
Assume $a>0$ and $a_n \geq 0$. how to verify that
$$\sum_{n=1}^{\infty}\frac{a_n}{(a+S_n)^{3/2}}\leq \int_0^{\infty}\frac{1}{(a+x)^{3/2}}\mathrm{d}x$$
where $S_n = a_1+a_2+\cdots+a_n$
Thanks very much
AI: $$ \int_{S_{n-1}}^{S_{n}}\frac{dx}{ (a+x)^{\frac{3}{2}}} \geq \in... |
H: Rationalisation Problem
Demonstrate by rationalizing the denominator that:
$$
\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}} = \frac{(\sqrt{a}+\sqrt{b}-\sqrt{c})(a+b-c-2\sqrt{ab})}{a^2 + b^2 + c^2 - 2(ab+ac+bc)}
$$
AI: $$
\frac{1}{\sqrt{a}+\sqrt{b}+\sqrt{c}} =\frac{\sqrt a+\sqrt b-\sqrt c}{(\sqrt{a}+\sqrt{b}+\sqrt{c... |
H: Strong inductive proof for this inequality using the Fibonacci sequence.
Problem
I need to determine for what natural numbers is $2n < F_n$, where $F_n$ is the $n^{th}$ Fibonacci number determined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1}+F_{n-2}$. I then need to prove my findings through strong induction.
What I... |
H: Related rates problem
I'm learning single variable calculus; I finished a section on related rates several weeks ago. I'm sure the novelty of related rates and simple optimization problems will wear off eventually, but right now I'm having a lot of fun solving these kinds of problems and creating my own. My questio... |
H: What do we mean when we say the Expected Value E[X] is linear?
I know that $E[2-X]$ for instance is equal to: $2 - E[X]$. And it makes perfect sense to me, because $f(x)=x$ is linear.
However, $E[X^2]$ is equal to $\sum_{j=1}^n$ $x^2f(x)$, and $f(x) = x^2 $ is quadratic.
What do we mean by linear here, because I s... |
H: Can the factorial function be written as a sum?
I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
AI: This one is pretty important:
$$n! = \sum_{\sigma\in S_n} 1$$
Edit: As Arkamis explains, $S_n$ is the symmetric group on $n$ letters. Each $\sigma\in S_n... |
H: Differentiation of $x$ to the power of $y$ with respect to $x$
As the title suggests, I need to differentiate $x$ to the power of $y$ with respect to $x$. Not sure how to start. Do I need to take natural log on both sides?
That is: $\dfrac{d}{dx}x^y=?$
AI: We need $$\frac{d(x^y)}{dx}$$
One of the ways could be:
Let... |
H: Why if $\rho(I_{\mathfrak{X}} - YA)<1$ then $YA$ is invertible on the $R(YA)$?
I am reading an article where I am stucked at one point. Below is my problem.
Given that $\mathfrak{X}$ and $\mathfrak{Y}$ are Banach spaces.
$A:\mathfrak{X} \to \mathfrak{Y}$ and $Y:\mathfrak{Y}\to \mathfrak{X}$ are linear bounded ope... |
H: Computing the expected value $E[X(X+5)]$
I know $ E[XY] = \int \int x y f(x,y) dx dy $ where $f(x,y) = f(x)f(y)$
But I am not entirely sure how to compute $E[X(X+5)]$.
Is it $\int f(x)(5 + \int f(x) dx) dx$ ?
AI: $$\mathbb{E}[X(X+5)]= \mathbb{E}[X^2+5X] = \int_{-\infty}^{\infty} (x^2+5x)f_X(x)dx$$
Also note that... |
H: Is there a name for this type of logical fallacy?
Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$.
Example: As $3$ is odd, $3$ is prime.
In this case, it is true that $3$ is odd, and that $3$ is prime, but the implication is false. If $9$ had been used ... |
H: Seeking clarity regarding normal subgroup
If $A$ is a normal subgroup of $B$ then is it required for $A$ and $B$ to be groups under the binary operation multiplication? what if they are just groups under the binary operation addition, can there still exist normal subgroup?
Like, set of Rational numbers is a group u... |
H: Check whether the three vectors $A(2,-1,2),B(1,2,-3),C(3,-4,7) $ are in the same plane
I want to check if three vectors are in the same plane, the vectors being
$$A(2,-1,2),B(1,2,-3),C(3,-4,7). $$
What I did so far is to create vector $AB ( -1,3,-5)$ and build the plane equation with the point $A$
$$-1(x-2)+3(y... |
H: Proof of a regular parallelogram
Given any figure with four vertices and four straight edges, prove that one can construct a perfect parallelogram by connecting the midpoints of such figure.
This to me is a very fundamental and interesting geometry problem.
How would I begin to prove this?
AI: We do it for a conv... |
H: ZF construction of the Kleene plus
Given a non-empty set $A$, a (non-empty) string of $A$ is a tuple $(a_1,a_2,...,a_n) \in A^n$, where $a_j \in A$, $\forall j \in \{ 1,2,...,n \}$, for some $n \in \mathbb{N}^*$.
The Kleene plus of $A$, informally, is a set $\displaystyle A^{+} = \bigcup_{n=1}^{\infty} A^n$. It is ... |
H: Closed and exact.
I tried this question, but I have no idea if I got it correctly.
On $\mathbb{R}^2$, let $\omega = (\sin^4 \pi x + \sin^2 \pi(x + y))dx - \cos^2 \pi(x + y)dy$.
Let $\eta$ be the unique $1$-form on the torus $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$
such that $p^* \eta = \omega,$ where
$p: \mathbb{R}... |
H: Find the point of intersection of the straight line $\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$ and plane $3X+8Y-9Z=0$
Find the point of intersection of the straight line
$$\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$$ and plane $3X+8Y-9Z=0$
the point of the line is $M(-1,2,-6)$ and direction vector of the line i... |
H: Prove that $\sqrt 5$ is irrational
I have to prove that $\sqrt 5$ is irrational.
Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common factor other than 1,
$$\frac{p}{q} = \sqrt5$$
$$\Rightarrow \frac{p^2}{q^2} = 5$$
... |
H: Cyclic subgroup of a quotient group
I encountered this question in a grad-level exam. I hope somebody could help me with this.
We have to choose one option.
Consider the group $\;G=\Bbb Q/\Bbb Z\;$ where $\Bbb Q$ and $\Bbb Z$ are the groups of rational numbers and integers respectively. Let $n$ be a positive intege... |
H: Difference between $R[c_1,c_2,\dots, c_n]$ and a finitely generated $R$-algebra.
What is the difference between $R[c_1,c_2,\dots, c_n]$ ($c_1, c_2,\dots, c_n\notin R$), where $R$ is a ring, and a finitely generated $R$-algebra?
Is the difference that if $c_1, c_2,\dots, c_n$ are the generators of the $R$-algebra,... |
H: $\sum_{n=1}^{\infty}f(z^n)$ converges uniformly with $f$ holomorphic
Let $f$ be an holomorphic function on the unit ball with $f(0)=0$.
Prove that $\sum_{n=1}^{\infty}f(z^n)$ is uniformly locally
convergent in the unit ball.
My attemp:
It is suffice to prove that $\sum_{n=1}^{\infty}f(z^n)$ converges uniforml... |
H: Approximations of fixed points of tangent.
This question comes from an exam, years ago.
Show that $f(x)=\tan x-x$, for every positive integer $n$, has exactly one root $x_n$ in the interval $(n\pi,n\pi+\pi/2)$. And show that $$x_n=n\pi+\frac{\pi}{2}-\frac{1}{n\pi}+\text{o}\left(\frac{1}{n}\right).$$
I can pro... |
H: Finite and infinite sets, cardinality question
Suppose there are infinite sets $A$, $B$ and $C$ such that $$|A| = |B| = |C| = |\mathbb{N}|\\ |D| = |\mathbb{R}|$$ and the finite set $E$
Give an example for the following (using the sets above). In case it's not possible, show why.
$(A \setminus D = B) \wedge (A \cap... |
H: Find the projection of the point on the plane
I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.
Do I need to use Gram–Schmidt process? If yes, is this the right formula?
$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$
What will the result be, ve... |
H: Simple question concerning the properties of the fundamental group
I need to prove that every element of the fundamental group has an inverse.
First we define a map $\phi:I\to I$ homotopic to $\operatorname{Id}_I$. If $\phi$ is the constant zero function isn't it true that $f\circ \phi \simeq f$? So if we denote th... |
H: Does "monotonic sequence" always mean "a sequence of real numbers"
When we say a sequence is monotonic, does that imply the sequence is Real Number Sequence? And other propositions about monotonic, all real-valued?
When I see some mathematical analysis books, sometimes they talk about some properties/facts like con... |
H: $\xi$ is the least upper bound of $M$.
$M$ is a set with upper bound.
Should set $M$ be an ordered set? or by deafult it is an ordered set, since it maybe an sub set of $R$. When we say a subset $M$ of $R$(ordered set), is $M$ also an ordered set?
a cut $S=(\xi )$ is denoted as the set made up of the left part ... |
H: Is there any way to prove it directly?
I'm trying to prove the following result:
In a first countable $T_1$ space $X$ for $E\subset X,~x\in X$ is an adherent point of $E\iff~\exists~(x_n)_n\in E$ such that $x_n\to x.$
When I'm considering the $E=\emptyset,$ I can't prove it directly without using $$\{\text{if}~~p... |
H: Is there any good strategy for computing null space of a matrix with entries $\cos x$ and $\sin x$?
For example, say
$A= \left ( \begin{matrix} \cos x & -\sin x & 0 \\ \cos y \sin x & \cos x \cos y & -\sin y \\ \sin x \sin y & \sin y \cos x & \cos y \end{matrix} \right)$.
How do i conpute null space of $A-I$?
Sinc... |
H: help deriving a closed formula for this "magic function"
I'm having trouble coming up with a closed formula for $n$ from the sequence of numbers generated by this function:
The following mystery function $M : N \times N \rightarrow N $ is defined by:
$$
M(m,n) =
\begin{cases}
m & n < 2m +1 \\
M(m+1, n-2m-1) &n \ge... |
H: Proving that the mothersequence converges to $x$ if any subsequence contains a subsequence which converges to $x$
Dear reader of this post,
I am currently working on some problems about sequences and their subsequences. I proved a claim and because this prove involves some elementary concepts, I would like to ask ... |
H: is $I^2=I$ true?
Suppose $I$ is an ideal of a ring with $1$. I think that $II=I^2=I$ but I am stuck showing it. I can easily show that $I^2\subseteq I$, but I dont know how to show that $I\subseteq I^2$. So is it actually true? If yes, how can I show it?
Definition of $I^2=\{\sum_{k=1}^m x_k y_k: m \in \mathbb{Z}_{... |
H: How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$?
I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.
The problem is I don't quite like the solution, as it depe... |
H: The relation between these two kinds $\text{Mod}$
What's the relation between these two kinds $\text{Mod}$
$M$ is a subset of integer set $Z$, $M$ is a $\text{Mod}$ if $\forall x,y\in Z$, we have
$$\begin{align*}u,v\in M \Longrightarrow x u+y v\in M\tag{1}\end{align*}$$
Can this kind Mod be extended? here, we are ... |
H: Similar matrices properties
So I have a question which I can not solve.
Assuming $A,B \in \mathbb{M_{n}(\mathbb{R})}$, $A$ similar to $B$, is it possible that $\det(A) = \det(B^{2})+1$?
We know that there exists $P$ (invertible) such that $P^{-1}AP=B$ and therefore $B^{2} = P^{-1}A^{2}P$. This means that
$$
\det(B... |
H: How to write $K$ as sum of $N$ integers?
How to write integer $K$ as sum of $N$ positive integers with minimum variance?
Obviously when $N|K$ the solution is each of integers being $\frac KN$ and the variance would be zero. But how about when this not the case?
I know that the answer is some of them being $\lfloor\... |
H: If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?
This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to composite numbers, or in other words: "... |
H: Why does $(a_n)$ bounded imply that $(b_n)$ is decreasing?
Why does $(a_n)$ bounded imply that $(b_n)$ is decreasing?
$$(a_n)=a_1,a_2,\dots\tag{1}$$
$$b_n=\sup (a_n,a_{n+1},\dots), c_n=\inf (a_n,a_{n+1},\dots)$$
If $\left(a_n\right)$ is bounded, then $\left(b_n\right)$ exists and $(b_n)$ is decreasing, $(c_n)$ is ... |
H: Convergence of Improper Integrals2
Test the convergence of $$\int_0^{\pi/2}\frac{\sin x}{x^n}\,dx$$
I tried doing it by comparison test by taking $\phi(x)=\dfrac{1}{x^n}$. Then
$$\lim_{n\rightarrow 0}\frac{f(x)}{\phi(x)}=\lim_{n\rightarrow 0}\sin x=0$$ This implies that if $\phi(x)$ converges then $f(x)$ also conve... |
H: Denseness of the set $\{f: \int_0^1 x^\alpha f''(x) dx = \int_0^1 x^\beta f''(x) dx = 0 \}$ in $C[0,1]$
Let $\alpha, \beta \in (-1,1) \setminus \{ 0 \}$. Is it true that the set
$$
\left\{f \in C^2[0,1]: \int_0^1 x^\alpha f''(x) dx = \int_0^1 x^\beta f''(x) dx = 0 \right\}
$$
is dense in $C[0,1]$? I think it is not... |
H: Intuitive reason for why many complex integrals vanish when the path is "blown-up"?
It is a standard trick for evaluating difficult integrals along the real line to consider a closed-contour and "blow-up" the complex part till it vanishes, leaving us with the residues picked up along the way. This is usually done b... |
H: Check the convergence $\sum_{n=1}^{\infty}\frac{\sin^3(n^2+11)}{n^4}$
I`m trying to check the convergence of this series but I don't know how to start.
$$\sum_{n=1}^{\infty}\frac{\sin^3(n^2+11)}{n^4}$$
I thought about using Comparison test. so I know when I have some $\theta$ any trigonometric function I will take ... |
H: What do we know about the distribution of Mersenne primes?
Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how far apart successive Mersenne primes can be. For example, is $M_{n+1} \le O((M_n)^e)$? Or, is $M_{n+1}$ always less than some power of $M_n$? If not, how close together can succes... |
H: Differentiating terms involving evaluation operator and Wronski matrix
We have the initial value problem $$\dot{\mathbf{y}} = f(\mathbf{y}), \mathbf{y}(0) = \mathbf{y_0},$$ with $f(\mathbf{y})$ continuously differentiable. There exists a $T > 0$ such that $\mathbf{y_0} = \mathbf{\Phi}^T \mathbf{y_0}$ and $\mathbf{y... |
H: How many squares are there modulo a Mersenne prime?
Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how many distinct natural numbers result from squaring the naturals modulo $M_n$.
As an example, $M_3 = 7$. If we take the naturals less than seven, we get:
$$1^2 \equiv 1 \bmod 7$$
$$2^2 \equ... |
H: When are $3$ vectors associative in triple cross products?
The question I am trying to show under what conditions
$$\vec{A}\times(\vec{B}\times\vec{C}) = (\vec{A}\times\vec{B})\times\vec{C}.$$
I have found that right-hand side of the above equation is equal to
\begin{align}
(\vec{A}\times\vec{B})\times\vec{C}
&=-\v... |
H: Completing metric space
In the completion of a metric space, a distance is defined on the set of equivalence classes of Cauchy sequences:
$$
\begin{align}
\tilde d:\tilde X\times \tilde X &\to \mathbb{R^+}\\
([x_n],[y_n]) &\mapsto \lim_{n\to \infty}(d(x_n,y_n))
\end{align}$$
with $x_n,y_n$ Cauchy sequences in the ... |
H: Could you please explain How to expand $(1 - \frac1x)^{-n}$ into a sum of powers of $x$?
Could you please explain how to expand $(1 - \frac1x)^{-n}$ into a sum of powers of $x$?
Thank you in advance.
AI: Consider the Taylor expansion of $(1-1/x)^{-n}$ about $1/x=0$:
$$\left ( 1-\frac{1}{x}\right)^{-n} = 1+ (-n) \l... |
H: Parametrization of unit sphere in $\mathbb{R}^3$
I would like to show (I'm not yet sure if it's true, though), that any vector $v\in \mathbb{R}^3$ with $\|v\| = 1$ can be written as $\left(\cos(\beta)\sin(\alpha),\; \sin(\alpha)\sin(\beta), \; \cos^2\left(\frac{\alpha}{2}\right)-\sin^2\left(\frac{\alpha}{2}\right)\... |
H: $n$ and $n^5$ have the same units digit?
Studying GCD, I got a question that begs to show that $n$ and $n^5$ has the same units digit ...
What would be an idea to be able to initiate such a statement?
testing
$0$ and $0^5=0$
$1$ and $1^5=1$
$2$ and $2^5=32$
In my studies, I have not got "mod", please use other mean... |
H: If $f<1$, $f(0)^2 + f'(0)^2=4$, exists $x_0$ s.t. $f''(x_0) + f(x_0)=0$
Suppose $f:\mathbb{R}\to\mathbb{R}$ is $C^2$, $f < 1$ for all $x$, and $f(0)^2 + f'(0)^2=4$. Show that $\exists x_0$ s.t. $f''(x_0) + f(x_0)=0$.
So far, I have let $\phi(x) = f(x)^2 + f'(x)^2$. Then $$\phi'(x) = 2f(x)f'(x) + 2f'(x)f''(x) = 2f'(... |
H: Approximate measures of sets with measures of borel subsets.
Show that for each subset $A$ of $\mathbb{R}$ there is a Borel subset of $B$ of $\mathbb{R}$ that includes $A$ such that $ \lambda (B) = \lambda ^*(A)$
If A is Borel it is evident? So we need to approximate A with some Borel subset, which is just a "litt... |
H: Why is this limit $\frac{e^x}{x^{x-1}}$coming out wrong?
Attempting to answer this question, I thought to evaluate the limit by taking the logarithm and then using L'Hopital's rule:
$$\begin{align}
L&=\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}\\
\ln{L}&=\lim_{x\to\infty}\frac{x}{(x-1)\ln(x)} \\
\ln{L}&=\lim_{x\to\infty}... |
H: Do we have $x^TDAx\ge \min(\lambda_D)\min(\lambda_A)x^Tx$ if $A$ is PD and D is both diagonal and PD?
Suppose matrix $A\in\mathbb{R}^{n\times n}$ is symmetric positive definite and $D\in\mathbb{R}^{n\times n}$ is both diagonal and positive definite, do we have the following result?
$$x^TDAx\ge \min(\lambda_D)\min(\... |
H: Evaluate $\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2} $ where $\gcd(m,n)=1$
i have no clue on how to evaluate:
$$\sum\limits_{(m,n) \in D,m < n} \frac{1}{n^2 m^2} \text{ where }D = \{ (m,n) \in (\mathbb{N}^*)^2 \mid \gcd(m,n) = 1\} $$
If someone is able to give me a hint...
Thanks much in advance.
AI: Call yo... |
H: Solving a straightforward linear ODE
I'm trying to solve this linear ODE, but I seem to get it wrong for some reason.
$$\dfrac{dy}{dt}-\dfrac{1}{2}y(t)=2\cos t.$$
I get
$$y(t)=e^{t/2}(c+2 \sin t).$$
How can this be wrong? And yet it seems to be.
AI: Hints:
Solve for the homogenous, $y_h = c_1e^{t/2}$ (you got tha... |
H: A definition of algebraic expression
Definition of algebraic expression
An algebraic expression is a collection of symbols; it may consist of one or more than one terms separated by either a $+$ or $-$ sign.
If by symbol we only mean letters such as $a, b$ or $c$, then what about this algebraic expression which con... |
H: Showing absolute convergence for series representation of $\frac{z}{exp(z) - 1}$
I would appreciate help (self-study) in showing that the related power series of
$\frac{z}{exp(z) - 1}$ converges absolutely for $|z| < 2\pi$ without looking at the specific terms of the series.
(I do know the series, other than the fi... |
H: The closure $\mathbb{Q}$
Why the closure $\mathbb{Q}$ is not itself? Since each ball around $q \in \mathbb{Q}$ contains a point in $\mathbb{Q}$ and a point in $\mathbb{R} \setminus \mathbb{Q}$.
AI: Take the sequence $a_n$ where $$a_n=\sum_{k=0}^n \frac{1}{n!}$$ Each of $a_n$ is in $\mathbb{Q}$ but the limit of $a_n... |
H: Is there a slowest rate of divergence of a series?
$$f(n)=\sum_{i=1}^n\frac{1}{i}$$
diverges slower than
$$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$
, by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow \infty}(f(n)-\ln(n))=\gamma$, so ... |
H: not both $2^n-1,2^n+1$ can be prime.
I am trying to prove that not both integers $2^n-1,2^n+1$ can be prime for $n \not=2$. But I am not sure if my proof is correct or not:
Suppose both $2^n-1,2^n+1$ are prime, then $(2^n-1)(2^n+1)=4^n-1$ has precisely two prime factors. Now $4^n-1=(4-1)(4^{n-1}+4^{n-2}+ \cdots +1)... |
H: Prove that $\sqrt 2 + \sqrt 3$ is irrational
I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?
Secondly, the book want... |
H: Properties of the Fourier transform of a certain function
In my research I met the Fourier transform of the function $f(x)=(1+x^2)^{-1/2}$. I was not able to find its explicit formula. Is this a function known as a 'special function'? I would like to know if it is nonnegative, summable, etc.
AI: If the fourier tran... |
H: Solving the differential equation $\frac{dC}{dt} = -\alpha C$
I am trying to solve the following problem:
After drinking a cup of coffee, the amount $C$ of caffeine in a person's
body obeys the differential equation
$$\frac{dC}{dt} = -\alpha C$$
where the constant $-\alpha$ has an approximate value of $0.14$ ... |
H: Proof that if $n
Can I get a proof of the fact that if $n<k$ and $A$ is an $n\times k$ matrix, then $A^{T}A$ is not invertible?
AI: Hint $rank(A) \leq n$ then $A^TA$ is an $k \times k$ matrix and
$$rank(A^TA) \leq rank(A) \leq n <k \,.$$ |
H: How do I determine the curvature of an arc length parameterized curve in the $xy$-plane?
I have a 2D curve in the $xy$-plane, which was arc length parameterized numerically, and fitted by cubic splines for both $x$ and $y$.
If one of the segments of the cubic spline is:
\begin{align}
x&=a_1s^3 + a_2s^2 + a_3s + a_4... |
H: When does this fact involving Lagrange's Theorem hold?
Let $G$ be a finite group, with subgroups $H$ and $K$, and $H \subseteq K \subseteq G$. Then we have
\begin{equation*}
[G\,:\,H] = [G\,:\,K][K\,:\,H].
\end{equation*}
Question: Do we require that $H\subseteq K$ be true for this property to hold, or is it true f... |
H: How can you fold a rectangular piece of paper once to get an 8 sided polygon?
If you start with a rectangular piece of paper, how can you fold it once to get an 8 sided polygon?
I am sure the solution is straightforward, but after trying some possibilities, I was unable to come up with such a construction. Could so... |
H: Am I going correctly?
Let $f(x)$ be a polynomial in $x$ and let $a, b$ be two real numbers where $a \neq b$
Show that if $f(x)$ is divided by $(x-a)(x-b)$ then the remainder is
$\frac{(x-a) f(b)-(x-b) f(a)}{b-a}$
MY APPROACH:-
Let $Q(x)$ be quotient so that :
$(x-a)(x-b)Q(x)+{ Remainder }=f(x)$
L.H.S, $(x-a)(x-b) \... |
H: Is $\frac{\cos(\frac{1}{z})}{z^2}$ meromorphic or Not?
my professor used the Cauchy Residue Theorem to evaluate the path integral (along the positively-oriented unit circle about the origin with winding number 3)
$$\int_{\gamma}\frac{\cos(\frac{1}{z})}{z^2}$$
His reasoning is that, when expanded into series,
$$\o... |
H: how would you prove that polynomial functions are not exponential?
here is one proof that I know but I am not totally sure if it is acceptable-
exponential functions are exponential: no matter how many times you differentiate them
e.g-
f(x)=e^x
first derivative f`(x)= e^x
2nd derivative f``(x)= e^x
3rd derivative ... |
H: Show that $23a^2$ is not the sum of 3 squares.
I know that Legendre's theorem states that a number is expressible as a sum of 3 squares iff. it's not of the form $4^x (8m+7)$, so I need to show that $23a^2$ is of this form, how could I go about doing this?
AI: Note that $4^x(8m + 7)$ is a product of two terms:
a p... |
H: How to determine the limits of $\cos(x/3) / \cos(x)$ over the domain $x \in [0, \pi/2]$?
I'm trying to determine the range of $y = \frac{\cos(x / 3)}{\cos(x)}$ over the domain $x \in [0, \pi / 2]$.
This is my attempt:
$$
\lim_{x \rightarrow 0} \frac{\cos(x / 3)}{\cos(x)} = \frac{\cos(0 / 3)}{\cos(0)} = 1
$$
and
$$
... |
H: How can I represent the $-3 Re(z) - 6 Im(z) \geq -2$ and $Re(z) > 2$ on a complex number plane?
How can I represent the $-3 Re(z) - 6 Im(z) \geq -2$ and $Re(z) > 2$ on a complex number plane?
Is it just $-3 - 6 * i \geq -2$ and $x >2$?
AI: On the complex plane, a plot of $x$ and $y$ is made where the complex number... |
H: How many options there are for $n$ people to shake hands exactly $r$ times?
Find how many options there are for $n$ people to shake hands exactly $r$ times while:
The same pair of people can't shake hands more than once
Order of hand shakes does not matter
So the solution I thought about is ordering all peopl... |
H: For a given symmetric and positive definite matrix M, find matrix C which fulfills CC^T = M and C^TC = D
D is the diagonal matrix containing the eigenvalues of M. Is it solvable and if so, how?
Thanks in advance!
AI: If $C$ is a square matrix, the equation is always solvable. Let $M=QDQ^T$ be an orthogonal diagonal... |
H: Eliminating $y$ from the system $cx − sy = 2$ and $sx + cy = 1$, where $c=\cos\theta$, $s=\sin\theta$
We will write $c = \cos\theta$ and $s = \sin\theta$ for ease of notation. Eliminate $y$ from the simultaneous equations
$$\begin{align}
cx − sy = 2 \\
sx + cy = 1
\end{align}$$
How could you eliminate $y$ from ... |
H: Converse to a proposition on homogeneous polynomials
I know that for a homogeneous polynomial $P$, if $P(x_1, ... , x_n) = 0$, then $P(ax_1, ..., ax_n) = 0$ for every $a$ in the field of $P$. Is the converse of this proposition true? That is, if $P(x_1, ... , x_n) =0$ implies $P(ax_1, ..., ax_n) = 0$ for every $a$ ... |
H: Integration by parts for evaluating a differential equation
Hi I am trying to evaluate a integral and I am hoping for some assistance
$$\int \frac{9x^2}{(x^6+9)} dx $$
I have rewritten the problem as shown below
$$\int 9x^2(x^6+9)^{-1}dx$$
Therefore I attempted the question by using integration by parts
$$\int udv ... |
H: Prove that $\lim_{n \to \infty}\frac{\sin\left(\frac{1}{\sqrt{n+1}+\sqrt{n}}\right)}{\frac{1}{\sqrt{n}}} = L, \quad L \in R$
In a question the answers say that:
$$
\lim_{n \to \infty}\frac{\sin\left(\frac{1}{\sqrt{n+1}+\sqrt{n}}\right)}{\frac{1}{\sqrt{n}}} = L, L \in R
$$
How?
AI: Multiply numerator and denominato... |
H: Uniform convergence of $f_n(z)=nz^n$ in the set $|z|<\frac{1}{2}$
In an exercise I have to prove that $f_n(z)=nz^n$ converges uniformly for $|z|<\frac{1}{2}$.
So I have to prove that:
$$\forall \varepsilon>0, \exists N \in \mathbb{N}:|nz^n-f(z)|<\varepsilon\ \ \ \text{if } n\geq N$$
My question is, how can I find t... |
H: Is this a characterization of the resolvent?
I am trying to understand a statement that is in some notes that I am reading right now. It is the following.
"Let $T$ be a bounded, self-adjoint operator, $\eta\in\mathbb{R}, \eta\neq 0$ and let $H$ be an Hilbert space. It can easily proved that $(T-i\eta)^{-1}$ is boun... |
H: Show that $(x_n)^{\infty}_{n=1}$ converges.
Let $(X, d)$ a complete metric space and $(x_n)^{\infty}_{n=1}$ a sequence such that $d(x_{n+1},x_n) \leq \alpha d(x_n, x_{n-1})$ for some $0<\alpha<1$ and for all $n\geq 2$. Show that $(x_n)^{\infty}_{n=1}$ converges.
Been dealing with this problem without success, any ... |
H: How many ways can we choose a team of $16$ people with $1$ leader and $4$ deputies out of $75$ people?
How many ways can we choose a team of 16 people with 1 leader and 4 deputies out of 75 people?
I figured that we can select the 16 people out of the 75 simply with $75\choose 16$ but then should i continue multipl... |
H: Prove that $\sum (-1)^n \sin(\sqrt{n+1}-\sqrt{n})$ converges by Leibniz
Prove that $\sum_{n=1}^{\infty}(-1)^n\sin\left(\sqrt{n+1}-\sqrt{n}\right)$ converges by Leibniz.
The answer says that $\sin$ is continuous, monotonic around $0$ and the limit there is $0$, therefore the series conditionally converges by the Lei... |
H: Geometry problem involving a cyclic quadrilateral and power of a point theorem?
Convex cyclic quadrilateral $ABCD$ are inscribed in circle $O$. $AB,CD$ intersect at $E$, $AD,BC$ intersect at $F$. Diagonals $AC, BD$ intersect at $X$. $M$ is midpoint of $EF$. $Y$ is midpoint of $XM$. circle $Y$ with diameter $XM$ int... |
H: Sum of two cubes equal to prime square
If $a,b\in \mathbb{N}$ find all primes $p$ such that $a^3+b^3=p^2$
My approach-
$a^3+b^3=(a+b)(a^2-ab+b^2)=p^2$ suppose $a+b=x$ and $a^2-ab+b^2=y$ then there are two cases- $(x,y)=(p^2,1),(p,p)$
Now I am struggling for case 02 where $(x,y)=(p,p)$
AI: In the case that $a+b=a^2-... |
H: Pullback Topology
Let $f:X\to Y$ be a bijection and $Y$ be a topological space. Let $T_X \triangleq \left\{
f^{-1}[U]:\, U \mbox{ open in Y}
\right\}$. Then is $T_X$ a topology on $X$ and if so, with it, is $X$ homeomorphic to $Y$?
AI: It is indeed a topology:
$X,\emptyset \in T$, since $f^{-1}(\emptyset)=\empty... |
H: Splitting field $L$ of polynomial $f \in K[x]$ with degree $n$ satisfies $[L:K] | n!$
Suppose $f \in K[x]$ is a polynomial with degree $n$, $f = (x-\alpha_1)...(x-\alpha_n)$ over the algebraic colsure. Let $L=K(\alpha_1,...,\alpha_n)$ be the splitting field of $f$. Prove that $[L:K]$ divides $n!$. I was able to pro... |
H: Examples of no-zero-measure meagre set
I know cantor set and rational numbers in $\mathbb{R}$ are meagre. But they are all zero measure.
So is there any meagre set that is non-zero measure?
AI: You should read about Fat Cantor sets - they are nowhere dense but have positive measure. |
H: The module $\text{Hom}_C(E,F)$ of two finitely generated projective $C$-modules
Let $C$ be an abelian ring and $E,F$ two finitely generated projective
modules. Then $\text{Hom}_C(E,F)$ is a finitely generated projective
$C$-module.
First of all, since $C$ is abelian, the abelian group $\text{Hom}_C(E,F)$ is a... |
H: Linear operator on $\ell^{\infty}$ is not surjective – well-definition of inverse operator
Let $T: \ell^{\infty} \rightarrow \ell^{\infty}$, such that $Tx = \left(\frac{1}{n} x_n\right)_{n\in \mathbb{N}}$. Claim: $T$ is not surjective.
I have been trying to come up with $y \in \ell^{\infty}$ such that there is no $... |
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